53 lines
1.4 KiB
Markdown
53 lines
1.4 KiB
Markdown
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---
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id: 5900f5241000cf542c510036
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title: 'Problem 437: Fibonacci primitive roots'
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challengeType: 5
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forumTopicId: 302108
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dashedName: problem-437-fibonacci-primitive-roots
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---
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# --description--
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When we calculate $8^n$ modulo 11 for $n = 0$ to 9 we get: 1, 8, 9, 6, 4, 10, 3, 2, 5, 7.
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As we see all possible values from 1 to 10 occur. So 8 is a primitive root of 11.
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But there is more:
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If we take a closer look we see:
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$$\begin{align} & 1 + 8 = 9 \\\\ & 8 + 9 = 17 ≡ 6\bmod 11 \\\\ & 9 + 6 = 15 ≡ 4\bmod 11 \\\\ & 6 + 4 = 10 \\\\ & 4 + 10 = 14 ≡ 3\bmod 11 \\\\ & 10 + 3 = 13 ≡ 2\bmod 11 \\\\ & 3 + 2 = 5 \\\\ & 2 + 5 = 7 \\\\ & 5 + 7 = 12 ≡ 1\bmod 11. \end{align}$$
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So the powers of 8 mod 11 are cyclic with period 10, and $8^n + 8^{n + 1} ≡ 8^{n + 2} (\text{mod } 11)$. 8 is called a Fibonacci primitive root of 11.
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Not every prime has a Fibonacci primitive root. There are 323 primes less than 10000 with one or more Fibonacci primitive roots and the sum of these primes is 1480491.
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Find the sum of the primes less than $100\\,000\\,000$ with at least one Fibonacci primitive root.
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# --hints--
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`fibonacciPrimitiveRoots()` should return `74204709657207`.
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```js
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assert.strictEqual(fibonacciPrimitiveRoots(), 74204709657207);
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```
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# --seed--
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## --seed-contents--
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```js
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function fibonacciPrimitiveRoots() {
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return true;
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}
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fibonacciPrimitiveRoots();
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```
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# --solutions--
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```js
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// solution required
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```
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